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Harmonics

October 29, 2012

Where Were We?

- We’re halfway through grading the mid-terms.
- For the next two weeks: more acoustics
- It’s going to get worse before it gets better
- Note: I’ve posted a supplementary reading on today’s lecture to the course web page.
- What have we learned?
- How to calculate the frequency of a periodic wave
- How to calculate the amplitude, RMS amplitude, and intensity of a complex wave
- Today: we’ll start putting frequency and intensity together

Complex Waves

- When more than one sinewave gets combined, they form a complex wave.
- At any given time, each wave will have some amplitude value.
- A1(t1) := Amplitude value of sinewave 1 at time 1
- A2(t1) := Amplitude value of sinewave 2 at time 1
- The amplitude value of the complex wave is the sum of these values.
- Ac(t1) = A1 (t1) + A2 (t1)

Complex Wave Example

- Take waveform 1:
- high amplitude
- low frequency

+

- Add waveform 2:
- low amplitude
- high frequency

=

- The sum is this complex waveform:

Greatest Common Denominator

- Combining sinewaves results in a complex periodic wave.
- This complex wave has a frequency which is the greatest common denominator of the frequencies of the component waves.
- Greatest common denominator = biggest number by which you can divide both frequencies and still come up with a whole number (integer).
- Example:
- Component Wave 1: 300 Hz
- Component Wave 2: 500 Hz
- Fundamental Frequency: 100 Hz

Why?

- Think: smallest common multiple of the periods of the component waves.
- Both component waves are periodic
- i.e., they repeat themselves in time.
- The pattern formed by combining these component waves...
- will only start repeating itself when both waves start repeating themselves at the same time.
- Example: 3 Hz sinewave + 5 Hz sinewave

For Example

- Starting from 0 seconds:
- A 3 Hz wave will repeat itself at .33 seconds, .66 seconds, 1 second, etc.
- A 5 Hz wave will repeat itself at .2 seconds, .4 seconds, .6 seconds, .8 seconds, 1 second, etc.
- Again: the pattern formed by combining these component waves...
- will only start repeating itself when they both start repeating themselves at the same time.
- i.e., at 1 second

Tidbits

- Important point:
- Each component wave will complete a whole number of periods within each period of the complex wave.
- Comprehension question:
- If we combine a 6 Hz wave with an 8 Hz wave...
- What should the frequency of the resulting complex wave be?

- To ask it another way:
- What would the period of the complex wave be?

Fourier’s Theorem

- Joseph Fourier (1768-1830)
- French mathematician
- Studied heat and periodic motion
- His idea:
- any complex periodic wave can be constructed out of a combination of different sinewaves.
- The sinusoidal (sinewave) components of a complex periodic wave = harmonics

The Dark Side

- Fourier’s theorem implies:
- sound may be split up into component frequencies...
- just like a prism splits light up into its component frequencies

Spectra

- One way to represent complex waves is with waveforms:
- y-axis: air pressure
- x-axis: time
- Another way to represent a complex wave is with a power spectrum (or spectrum, for short).
- Remember, each sinewave has two parameters:
- amplitude
- frequency
- A power spectrum shows:
- intensity (based on amplitude) on the y-axis
- frequency on the x-axis

Example

- Go to Praat
- Generate a complex wave with 300 Hz and 500 Hz components.
- Look at waveform and spectral views.
- And so on and so forth.

Fourier’s Theorem, part 2

- The component sinusoids (harmonics) of any complex periodic wave:
- all have a frequency that is an integer multiple of the fundamental frequency of the complex wave.
- This is equivalent to saying:
- all component waves complete an integer number of periods within each period of the complex wave.

Example

- Take a complex wave with a fundamental frequency of 100 Hz.
- Harmonic 1 = 100 Hz
- Harmonic 2 = 200 Hz
- Harmonic 3 = 300 Hz
- Harmonic 4 = 400 Hz
- Harmonic 5 = 500 Hz
- etc.

Deep Thought Time

- What are the harmonics of a complex wave with a fundamental frequency of 440 Hz?
- Harmonic 1: 440 Hz
- Harmonic 2: 880 Hz
- Harmonic 3: 1320 Hz
- Harmonic 4: 1760 Hz
- Harmonic 5: 2200 Hz
- etc.

- For complex waves, the frequencies of the harmonics will always depend on the fundamental frequency of the wave.

A new spectrum

- Sawtooth wave at 440 Hz
- Also check out a sawtooth wave at 150 Hz

Another Representation

- Spectrograms
- = a three-dimensional view of sound
- Incorporate the same dimensions that are found in spectra:
- Intensity (on the z-axis)
- Frequency (on the y-axis)
- And add another:
- Time (on the x-axis)
- Back to Praat, to generate a complex tone with component frequencies of 300 Hz, 2100 Hz, and 3300 Hz

Something Like Speech

- Check this out:

- One of the characteristic features of speech sounds is that they exhibit spectral change over time.

source: http://www.haskins.yale.edu/featured/sws/swssentences/sentences.html

Harmonics and Speech

- Remember: trilling of the vocal folds creates a complex wave, with a fundamental frequency.
- This complex wave consists of a series of harmonics.
- The spacing between the harmonics--in frequency--depends on the rate at which the vocal folds are vibrating (F0).
- We can’t change the frequencies of the harmonics independently of each other.
- Q: How do we get the spectral changes we need for speech?

Resonance

- Answer: we change the intensity of the harmonics
- Listen to this:
- source: http://www.let.uu.nl/~audiufon/data/e_boventoon.html

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